Subgroups — C₇×#22804 #13200020

Order 1792.

710 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2943	C₂	1	1	yes	maximal, characteristic
2	C₇×#2925	C₂	1	2	yes	maximal, complemented
3	C₇×#1975	C₂	1	1	yes	maximal, characteristic
4	C₇×#1974	C₂	1	1	yes	maximal, characteristic
5	C₇×#1929	C₂	1	2	yes	maximal, complemented
6	C₇×#1929	C₂	1	2	yes	maximal, complemented
7	C₇×#1849	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1806	C₂	1	2	yes	maximal, complemented
9	C₇×#1773	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1771	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×#135	C₂²	1	1	yes	characteristic
13	C₄×C₇×#41	C₂²	1	2	yes	—
14	C₇×#442	C₄	1	2	yes	complemented
15	C₇×#441	C₄	1	2	yes	complemented
16	C₇×#440	C₄	1	2	yes	complemented
17	C₇×#440	C₄	1	2	yes	complemented
18	C₇×#433	—	2	2	no	—
19	C₇×#433	—	2	2	no	—
20	C₇×#432	—	2	2	no	—
21	C₇×#431	—	2	2	no	—
22	C₂×C₇×#108	C₂²	1	1	yes	characteristic
23	C₂×C₇×#108	C₂²	1	1	yes	characteristic
24	C₂×C₇×#108	C₂²	1	2	yes	—
25	C₇×#427	C₂²	1	1	yes	characteristic
26	C₇×#423	C₂²	1	2	yes	—
27	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₇×#104	C₂²	1	2	yes	—
29	C₂×C₇×#104	C₂²	1	2	yes	—
30	C₇×#416	C₂²	1	2	yes	—
31	C₇×#408	C₂²	1	2	yes	—
32	C₇×#408	C₂²	1	2	yes	—
33	C₇×#405	C₂²	1	2	yes	—
34	C₇×#405	C₂²	1	2	yes	—
35	C₂×C₇×#97	C₂²	1	1	yes	characteristic
36	C₂×C₇×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
38	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
39	C₇×#339	C₂²	1	2	yes	—
40	C₇×#339	C₂²	1	2	yes	—
41	C₇×#336	C₂²	1	1	yes	characteristic
42	C₇×#336	C₂²	1	1	yes	characteristic
43	#22804	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×C₇×#41	C₂³	1	2	yes	—
45	C₇×#135	—	2	2	no	—
46	C₇×#135	—	2	2	no	—
47	C₂×C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₇×SD₁₆	C₂×C₄	1	2	yes	—
49	C₂×C₇×D₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₇×M₁₆	C₂³	1	2	yes	—
51	C₂×C₇×M₁₆	C₂³	1	2	yes	—
52	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
55	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
56	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
57	C₄×C₇×Q₈	C₂×C₄	1	2	yes	—
58	C₄×C₇×D₄	—	2	2	no	—
59	C₄×C₇×D₄	—	2	2	no	—
60	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
61	C₄×C₇×D₄	C₂×C₄	1	2	yes	—
62	C₇×#117	—	2	2	no	—
63	C₇×#117	C₂³	1	2	yes	—
64	C₂×C₇×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄²×C₇	—	2	2	no	—
66	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
69	C₇×#108	—	2	2	no	—
70	C₇×#108	—	2	2	no	—
71	C₇×#108	—	4	2	no	—
72	C₇×#107	C₂×C₄	1	1	yes	characteristic
73	C₇×#107	C₂×C₄	1	1	yes	characteristic
74	C₇×#106	C₂×C₄	1	2	yes	—
75	C₇×C₄⋊C₈	—	2	2	no	—
76	C₇×C₄⋊C₈	—	2	2	no	—
77	C₇×C₄⋊C₈	D₄	1	2	yes	—
78	C₇×C₄⋊C₈	D₄	1	2	yes	—
79	C₇×#104	—	4	2	no	—
80	C₇×#104	—	4	2	no	—
81	C₇×#103	C₂×C₄	1	2	yes	—
82	C₇×#103	C₂×C₄	1	2	yes	—
83	C₇×#102	C₂×C₄	1	2	yes	—
84	C₇×#102	C₂×C₄	1	2	yes	—
85	C₇×C₂²⋊C₈	—	2	2	no	—
86	C₇×C₂²⋊C₈	—	2	2	no	—
87	C₇×C₂²⋊C₈	—	2	2	no	—
88	C₇×C₂²⋊C₈	—	2	2	no	—
89	C₇×#97	—	2	2	no	—
90	C₇×#97	—	2	2	no	—
91	C₇×#97	—	2	2	no	—
92	C₇×#97	C₂×C₄	1	2	yes	—
93	C₇×#97	C₂×C₄	1	2	yes	—
94	C₄×C₇×C₈	—	2	2	no	—
95	C₄×C₇×C₈	—	2	2	no	—
96	C₄×C₇×C₈	—	2	2	no	—
97	C₇×#95	—	2	1	no	—
98	#2943	C₂×C₇	1	1	yes	characteristic
99	#2925	C₂×C₇	1	2	yes	complemented
100	#1975	C₂×C₇	1	1	yes	characteristic
101	#1974	C₂×C₇	1	1	yes	characteristic
102	#1929	C₂×C₇	1	2	yes	complemented
103	#1929	C₂×C₇	1	2	yes	complemented
104	#1849	C₂×C₇	1	1	yes	characteristic, complemented
105	#1806	C₂×C₇	1	2	yes	complemented
106	#1773	C₂×C₇	1	1	yes	characteristic, complemented
107	#1772	C₂×C₇	1	1	yes	characteristic, complemented
108	#1771	C₂×C₇	1	1	yes	characteristic, complemented
109	C₇×#41	—	2	2	no	—
110	C₇×#41	—	2	2	no	—
111	C₇×#41	—	2	2	no	—
112	C₇×#41	—	2	2	no	—
113	C₂×C₇×Q₈	C₂²×C₄	1	2	yes	—
114	C₂×C₇×D₄	—	2	2	no	—
115	C₂×C₇×D₄	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₇	—	2	1	no	—
117	C₂²×C₄×C₇	—	2	2	no	—
118	C₂²×C₄×C₇	—	2	2	no	—
119	C₂²×C₄×C₇	—	2	2	no	—
120	C₂²×C₄×C₇	—	2	1	no	—
121	C₂²×C₄×C₇	#41	1	1	yes	characteristic
122	C₂²×C₄×C₇	#41	1	1	yes	characteristic
123	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
124	C₇×Q₁₆	—	2	2	no	—
125	C₇×SD₁₆	—	2	4	no	—
126	C₇×D₈	—	2	2	no	—
127	C₇×M₁₆	—	2	2	no	—
128	C₇×M₁₆	—	2	2	no	—
129	C₇×M₁₆	—	2	2	no	—
130	C₇×M₁₆	—	2	2	no	—
131	C₂×C₇×C₈	—	2	1	no	—
132	C₂×C₇×C₈	—	2	1	no	—
133	C₂×C₇×C₈	—	2	1	no	—
134	C₂×C₇×C₈	—	2	1	no	—
135	C₂×C₇×C₈	—	2	1	no	—
136	C₂×C₇×C₈	—	2	1	no	—
137	C₂×C₇×C₈	—	2	2	no	—
138	C₂×C₇×C₈	—	2	2	no	—
139	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
140	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
144	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
145	C₂×C₇×C₈	#41	1	2	yes	—
146	C₂×C₇×C₈	#41	1	2	yes	—
147	C₇×C₄⋊C₄	—	2	2	no	—
148	C₇×C₄⋊C₄	—	2	2	no	—
149	C₇×C₄⋊C₄	—	4	2	no	—
150	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
151	C₇×C₄⋊C₄	C₂²×C₄	1	2	yes	—
152	C₇×C₂²⋊C₄	—	2	2	no	—
153	C₇×C₂²⋊C₄	—	2	2	no	—
154	C₇×C₂²⋊C₄	—	2	2	no	—
155	C₄²×C₇	—	2	2	no	—
156	C₄²×C₇	—	2	2	no	—
157	C₄²×C₇	—	2	2	no	—
158	C₄²×C₇	—	2	1	no	—
159	C₄²×C₇	—	2	1	no	—
160	C₄²×C₇	—	2	1	no	—
161	C₄²×C₇	—	2	1	no	—
162	C₄²×C₇	C₂²×C₄	1	2	yes	—
163	C₄²×C₇	C₂×D₄	1	2	yes	—
164	C₂×#135	C₂²×C₇	1	1	yes	characteristic
165	C₄×#41	C₂²×C₇	1	2	yes	—
166	#442	C₄×C₇	1	2	yes	complemented
167	#441	C₄×C₇	1	2	yes	complemented
168	#440	C₄×C₇	1	2	yes	complemented
169	#440	C₄×C₇	1	2	yes	complemented
170	#433	—	2	2	no	—
171	#433	—	2	2	no	—
172	#432	—	2	2	no	—
173	#431	—	2	2	no	—
174	C₂×#108	C₂²×C₇	1	1	yes	characteristic
175	C₂×#108	C₂²×C₇	1	1	yes	characteristic
176	C₂×#108	C₂²×C₇	1	2	yes	—
177	#427	C₂²×C₇	1	1	yes	characteristic
178	#423	C₂²×C₇	1	2	yes	—
179	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
180	C₂×#104	C₂²×C₇	1	2	yes	—
181	C₂×#104	C₂²×C₇	1	2	yes	—
182	#416	C₂²×C₇	1	2	yes	—
183	#408	C₂²×C₇	1	2	yes	—
184	#408	C₂²×C₇	1	2	yes	—
185	#405	C₂²×C₇	1	2	yes	—
186	#405	C₂²×C₇	1	2	yes	—
187	C₂×#97	C₂²×C₇	1	1	yes	characteristic
188	C₂×#97	C₂²×C₇	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
190	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
191	#339	C₂²×C₇	1	2	yes	—
192	#339	C₂²×C₇	1	2	yes	—
193	#336	C₂²×C₇	1	1	yes	characteristic
194	#336	C₂²×C₇	1	1	yes	characteristic
195	C₂³×C₇	—	2	2	no	—
196	C₂³×C₇	C₂×#41	1	1	yes	characteristic
197	C₇×Q₈	—	2	2	no	—
198	C₇×Q₈	—	2	2	no	—
199	C₇×D₄	—	2	2	no	—
200	C₇×D₄	—	2	2	no	—
201	C₇×D₄	—	4	2	no	—
202	C₇×D₄	—	4	2	no	—
203	C₂×C₄×C₇	—	2	2	no	—
204	C₂×C₄×C₇	—	2	2	no	—
205	C₂×C₄×C₇	—	2	2	no	—
206	C₂×C₄×C₇	—	2	2	no	—
207	C₂×C₄×C₇	—	4	1	no	—
208	C₂×C₄×C₇	—	4	1	no	—
209	C₂×C₄×C₇	—	4	4	no	—
210	C₂×C₄×C₇	—	4	1	no	—
211	C₂×C₄×C₇	—	4	1	no	—
212	C₂×C₄×C₇	—	4	2	no	—
213	C₂×C₄×C₇	—	4	2	no	—
214	C₂×C₄×C₇	—	4	1	no	—
215	C₂×C₄×C₇	—	4	1	no	—
216	C₂×C₄×C₇	—	4	1	no	—
217	C₂×C₄×C₇	—	8	2	no	—
218	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
219	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
220	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
226	C₇×C₈	—	2	2	no	—
227	C₇×C₈	—	2	1	no	—
228	C₇×C₈	—	2	1	no	—
229	C₇×C₈	—	2	2	no	—
230	C₇×C₈	—	4	2	no	—
231	C₇×C₈	—	4	2	no	—
232	C₇×C₈	#117	1	2	yes	—
233	C₇×C₈	#117	1	2	yes	—
234	C₂×#41	C₂³×C₇	1	2	yes	—
235	#135	—	2	2	no	—
236	#135	—	2	2	no	—
237	C₂×Q₁₆	C₂×C₄×C₇	1	1	yes	characteristic
238	C₂×SD₁₆	C₂×C₄×C₇	1	2	yes	—
239	C₂×D₈	C₂×C₄×C₇	1	1	yes	characteristic
240	C₂×M₁₆	C₂³×C₇	1	2	yes	—
241	C₂×M₁₆	C₂³×C₇	1	2	yes	—
242	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
243	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
244	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
245	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
246	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
247	C₄×Q₈	C₂×C₄×C₇	1	2	yes	—
248	C₄×D₄	—	2	2	no	—
249	C₄×D₄	—	2	2	no	—
250	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
251	C₄×D₄	C₂×C₄×C₇	1	2	yes	—
252	#117	—	2	2	no	—
253	#117	C₂³×C₇	1	2	yes	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄²	—	2	2	no	—
256	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
257	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
259	#108	—	2	2	no	—
260	#108	—	2	2	no	—
261	#108	—	4	2	no	—
262	#107	C₂×C₄×C₇	1	1	yes	characteristic
263	#107	C₂×C₄×C₇	1	1	yes	characteristic
264	#106	C₂×C₄×C₇	1	2	yes	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	2	no	—
267	C₄⋊C₈	C₇×D₄	1	2	yes	—
268	C₄⋊C₈	C₇×D₄	1	2	yes	—
269	#104	—	4	2	no	—
270	#104	—	4	2	no	—
271	#103	C₂×C₄×C₇	1	2	yes	—
272	#103	C₂×C₄×C₇	1	2	yes	—
273	#102	C₂×C₄×C₇	1	2	yes	—
274	#102	C₂×C₄×C₇	1	2	yes	—
275	C₂²⋊C₈	—	2	2	no	—
276	C₂²⋊C₈	—	2	2	no	—
277	C₂²⋊C₈	—	2	2	no	—
278	C₂²⋊C₈	—	2	2	no	—
279	#97	—	2	2	no	—
280	#97	—	2	2	no	—
281	#97	—	2	2	no	—
282	#97	C₂×C₄×C₇	1	2	yes	—
283	#97	C₂×C₄×C₇	1	2	yes	—
284	C₄×C₈	—	2	2	no	—
285	C₄×C₈	—	2	2	no	—
286	C₄×C₈	—	2	2	no	—
287	#95	—	2	1	no	—
288	C₂²×C₇	—	2	1	no	—
289	C₂²×C₇	—	2	1	no	—
290	C₂²×C₇	—	4	2	no	—
291	C₂²×C₇	—	4	2	no	—
292	C₂²×C₇	—	4	2	no	—
293	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
294	C₂²×C₇	#555	1	1	yes	characteristic
295	C₂²×C₇	#556	1	1	yes	characteristic
296	C₄×C₇	—	2	2	no	—
297	C₄×C₇	—	2	2	no	—
298	C₄×C₇	—	4	2	no	—
299	C₄×C₇	—	4	2	no	—
300	C₄×C₇	—	4	2	no	—
301	C₄×C₇	—	4	2	no	—
302	C₄×C₇	—	8	1	no	—
303	C₄×C₇	C₂×#117	1	1	yes	characteristic
304	C₄×C₇	#520	1	1	yes	characteristic
305	C₄×C₇	#547	1	1	yes	characteristic
306	C₄×C₇	#550	1	1	yes	characteristic
307	#41	—	2	2	no	—
308	#41	—	2	2	no	—
309	#41	—	2	2	no	—
310	#41	—	2	2	no	—
311	C₂×Q₈	C₂²×C₄×C₇	1	2	yes	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	C₂²×C₄×C₇	1	2	yes	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	C₇×#41	1	1	yes	characteristic
320	C₂²×C₄	C₇×#41	1	1	yes	characteristic
321	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
322	Q₁₆	—	2	2	no	—
323	SD₁₆	—	2	4	no	—
324	D₈	—	2	2	no	—
325	M₁₆	—	2	2	no	—
326	M₁₆	—	2	2	no	—
327	M₁₆	—	2	2	no	—
328	M₁₆	—	2	2	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
338	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
339	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
340	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
341	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
342	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
343	C₂×C₈	C₇×#41	1	2	yes	—
344	C₂×C₈	C₇×#41	1	2	yes	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	4	2	no	—
348	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
349	C₄⋊C₄	C₂²×C₄×C₇	1	2	yes	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	2	2	no	—
352	C₂²⋊C₄	—	2	2	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	1	no	—
357	C₄²	—	2	1	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	2	1	no	—
360	C₄²	C₂²×C₄×C₇	1	2	yes	—
361	C₄²	C₂×C₇×D₄	1	2	yes	—
362	C₂×C₇	—	2	1	no	—
363	C₂×C₇	—	2	1	no	—
364	C₂×C₇	—	8	2	no	—
365	C₂×C₇	#2354	1	1	yes	characteristic
366	C₂×C₇	#2955	1	1	yes	characteristic
367	C₂×C₇	#2956	1	1	yes	characteristic
368	C₂³	—	2	2	no	—
369	C₂³	C₂×C₇×#41	1	1	yes	characteristic
370	Q₈	—	2	2	no	—
371	Q₈	—	2	2	no	—
372	D₄	—	2	2	no	—
373	D₄	—	2	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	4	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
392	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
393	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
394	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
395	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
396	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
397	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
398	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
399	C₈	—	2	2	no	—
400	C₈	—	2	1	no	—
401	C₈	—	2	1	no	—
402	C₈	—	2	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₈	C₇×#117	1	2	yes	—
406	C₈	C₇×#117	1	2	yes	—
407	C₇	#22804	1	1	yes	characteristic, direct product factor, Sylow
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
414	C₂²	C₇×#555	1	1	yes	characteristic
415	C₂²	C₇×#556	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	2	no	—
422	C₄	—	8	1	no	—
423	C₄	C₂×C₇×#117	1	1	yes	characteristic
424	C₄	C₇×#520	1	1	yes	characteristic
425	C₄	C₇×#547	1	1	yes	characteristic
426	C₄	C₇×#550	1	1	yes	characteristic
427	C₂	—	2	1	no	—
428	C₂	—	2	1	no	—
429	C₂	—	8	2	no	—
430	C₂	C₇×#2354	1	1	yes	characteristic
431	C₂	C₇×#2955	1	1	yes	characteristic
432	C₂	C₇×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂×C₇×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₄⋊C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₇×#2925	C₂	semidirect
5. C₇×#1929	C₂	semidirect
6. C₇×#1929	C₂	semidirect
7. C₇×#1849	C₂	semidirect
8. C₇×#1806	C₂	semidirect
9. C₇×#1773	C₂	semidirect
10. C₇×#1772	C₂	semidirect
11. C₇×#1771	C₂	semidirect
14. C₇×#442	C₄	semidirect
15. C₇×#441	C₄	semidirect
16. C₇×#440	C₄	semidirect
17. C₇×#440	C₄	semidirect
43. #22804	C₇	direct (N × H)
99. #2925	C₂×C₇	semidirect
102. #1929	C₂×C₇	semidirect
103. #1929	C₂×C₇	semidirect
104. #1849	C₂×C₇	semidirect
105. #1806	C₂×C₇	semidirect
106. #1773	C₂×C₇	semidirect
107. #1772	C₂×C₇	semidirect
108. #1771	C₂×C₇	semidirect
166. #442	C₄×C₇	semidirect
167. #441	C₄×C₇	semidirect
168. #440	C₄×C₇	semidirect
169. #440	C₄×C₇	semidirect
407. C₇	#22804	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#2943	2
C₇×#2925	2
C₇×#1975	2
C₇×#1974	2
C₇×#1929	2
C₇×#1849	2
C₇×#1806	2
C₇×#1773	2
C₇×#1772	2
C₇×#1771	2
#22804	7

Subgroups — C7×#22804 #13200020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22804 #13200020